Complete Classification and Nondegeneracy of $N$-Component Cubic Nonlinear Schrödinger System in $\mathbb{R}$

We study the one-dimensional cubic nonlinear Schrödinger system $$u_i^{\mathrm{\prime }\mathrm{\prime }}+2\left(\sum _{k=1}^N{u}_k^2\right)u_i=-{\mu }_i{u}_{i}in\mathbb{R},i=1,2,\cdots ,N,$$ where $u=(u_1,\cdots ,u_N)\in (H^1(\mathbb{R}){)}^N$, ${\mu }_1\le {\mu }_2\le \cdots \le {\mu }_N<0$, and $N\ge 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1,2,\cdots ,N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt{\mathrm{\mid }{\mu }_i\mathrm{\mid }}$, which yields a complete characterization of normalized solutions satisfying ${\int }_{\mathbb{R}}{u}_i^{2}dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.